Please show work Let G be a graph with the node set V and ed

Please show work

Let G be a graph with the node set V and edge set E. Suppose that we are given weights W_ij greaterthanorequalto 0 for each edge (i, j) in the graph where w_ij. = w_ji. If (i, j) E then w_ij = 0. We want to determine a subset of the vertices S V that maximizes the sum of the weights over those edges that have one end point in S and the other end point in the complement S = V \\ S. Write an optimization model.

Solution

Definition (Independent Set). Given an undirected Graph G = (V,E) an independent set is a subset of nodes U V , such that no two nodes in U are adjacent. An independent set is maximal if no node can be added without violating independence. An independent set of maximum cardinality is called maximum1) a maximal independent set (MIS)

A node v joins the MIS in step 2 with probability p 1 4d(v) . Proof: Let M be the set of marked nodes in step 1. Let H(v) be the set of neighbors of v with higher degree, or same degree and higher identifier. Using independence of the random choices of v and nodes in H(v) in Step 1 we get P [v / MIS|v M] = P [w H(v), w M|v M] = P [w H(v), w M] ! wH(v) P [w M] = ! wH(v) 1 2d(w) ! wH(v) 1 2d(v) d(v) 2d(v) = 1 2